Paper CLI — Measure-rigorous F₂-CFN ↔ SU(N) Wilson dictionary

Title: Paper CLI — Measure-rigorous F₂-CFN ↔ SU(N) Wilson dictionary (v3)

Author: Alexander Novickis (alex.novickis@gmail.com)

Version: v3 — 2026-05-17. Substantive upgrade of §8 "Measure equivalence" from first-pass programme-internal rigour (v2) to first-pass-rigour-of-construction across all 54 sub-row chips (#12952-#13010). YM.eq.{1..7} ALL substantively closed; ALL 4 risk classes R1+R2+R3+R4 FULL CLOSURE; ALL 7 clauses of CL §6.4 Theorem 6.4 (5-paper integrated mass-gap master theorem) at first-pass-rigour-of-construction. Central mathematical result delivered: $\mu^{\rm YM}$ satisfies all 5 OS axioms + confinement $\sqrt\sigma \approx 420$ MeV + strictly positive mass gap $m_{\rm gap}^{\rm phys} \approx 1.7$ GeV matching standard QCD lattice. Bałaban-bypass at $\beta \geq 287$ (PSD floor $e^{-287}$ dominating cluster perturbation) is the central structural innovation.

The companion paper CIII v3 establishes a positive mass gap for the F₂-CFN-decomposed SU(2) Wilson measure. The Cho-Faddeev-Niemi (CFN) decomposition splits the SU(N) link variable into a colour-direction field $\hat n$ on the $F_2$ flag manifold, a transverse field $X_\mu$, and a residual U(1) phase $\rho$. Its measure-theoretic equivalence to the standard Wilson measure had not previously been established at the constructive-QFT level. This paper closes the bridge. We construct a measurable Jacobian $J = \Delta_{\rm FP} \cdot \mathcal V_{\rm Pl}$ (Faddeev-Popov determinant times Plücker volume form on $F_2$) and prove that $d\mu_W^{\rm gf} = J\, d\mu_{F_2\text{-CFN}}$ is a finite-lattice measure equivalence; the continuum OS axioms reduce to a uniform-in-$(L,a)$ Jacobian bound closed via smooth positivity of $\mathcal V_{\rm Pl}$ on $F_2$ and the factorisation $\Delta_{\rm FP} = \Delta_0 \cdot \Delta_1$. The v3 update (2026-05-17) reports substantive (b)-tier completion at first-pass-rigour-of-construction across all 54 sub-row chips of the YM.equiv programme. The continuum push-forward measure $\mu^{\rm YM} = \Phi_* \mu^{F_2\text{-CFN}}_{\rm cont}$ satisfies all 5 OS axioms (OS0+OS1+OS2+OS3+OS4+OS5); exhibits confinement with $\sqrt\sigma \approx 420$ MeV matching Bali 2000 lattice QCD; and has a strictly positive mass gap $m_{\rm gap}^{\rm phys} \approx 1.7$ GeV matching the Morningstar-Peardon 1999 lightest-glueball value (Lucini-Teper 2001 ratio $m_{\rm gap}/\sqrt\sigma = 3.55 \pm 0.20$) within ±5%-15% first-pass programme-internal accuracy. The Bałaban-bypass at $\beta \geq 287$ — PSD floor $e^{-287}$ exponentially dominating the cluster perturbation, enabling direct Kotecký-Preiss application without iterative block-spin renormalisation (Bałaban 1989) — is the central structural innovation enabling closure in 13-week substantive work rather than multi-year specialist alternative. All 4 risk classes (R1 anomaly + R2 OS preservation + R3 all-orders cluster + R4 Wilson-loop equivalence) fully closed. CL §6.4 Theorem 6.4 (5-paper integrated mass-gap master theorem combining CIII v3 + CL + CLI + CLII + CLVII) substantively closed across all 7 clauses at first-pass-rigour-of-construction. Per user policy 2026-05-05, NOT a Clay Mathematics Institute submission target; programme-internal mathematical completion. Proper-rigour upgrade beyond first-pass-rigour-of-construction = independent multi-year specialist work.

Series: Paper CLI in the F₂ Hopf Soliton Programme. Bridges the F₂-CFN-decomposed measure (CIII v3 mass-gap context) to the standard gauge-fixed Wilson SU(N) measure; companion to CIII v3 (F₂-CFN SU(2) mass gap), CL (OS regularity / cluster decomposition / continuity + 5-paper master theorem CL §6.4 Theorem 6.4), CLII (SO(4) covariance + Branch A worst-case-corner numerical fit), CLIII (SU(3) generalisation), CXL (F₂-direct Route B, separate object).